Updated for the 2026-2027 CFA® Level I curriculum.
A joint probability distribution connects each possible pair of asset returns with the probability that the pair will occur. You can use those probabilities to calculate expected returns, covariance, standard deviations, and correlation.
For CFA Level I, the calculation follows a clear sequence. Start with each asset’s expected return, measure how the paired returns differ from their means, and then convert covariance into a standardized correlation coefficient.
Quick Answer
To calculate covariance from a joint probability distribution, multiply each state’s probability by the product of the two assets’ return deviations, then add the results. Correlation standardizes that covariance by dividing it by the product of the assets’ standard deviations.
Key Takeaways About Covariance and Correlation Using Joint Probability
A joint probability represents the likelihood of one specific pair of returns occurring.
The probabilities across all possible paired outcomes must sum to 1.
You need both expected returns before you can calculate covariance.
Positive covariance means the returns generally move above or below their means together.
Negative covariance means one return tends to be above its mean when the other is below.
Correlation standardizes covariance to a range from 1 to +1.
Decimal and percentage inputs produce different covariance figures, so use one unit consistently.
What You Need to Know for CFA Level I
When working with covariance and correlation examples, you should be able to:
Read a joint probability distribution correctly.
Confirm that the state probabilities sum to 1.
Calculate the expected return of each asset.
Calculate covariance from probability-weighted cross-products.
Calculate each asset’s variance and standard deviation.
Calculate the correlation coefficient from covariance.
Interpret the sign and strength of the relationship.
Keep percentage and decimal units consistent throughout the calculation.
What Is a Joint Probability Distribution?
A joint probability distribution assigns a probability to each possible combination of outcomes for two assets or variables.
For example, one state may show that Asset A earns 12% while Asset B earns 10%, with a probability of 30%. The 30% probability applies to that paired outcome. It does not represent the standalone probability of either return.
A valid joint probability distribution follows two rules:
Every probability must fall between 0 and 1.
The probabilities of all possible paired outcomes must sum to 1.
The distribution gives you the information needed to calculate how the two assets behave together across different states.
Covariance and Correlation Formulas
Begin by calculating the expected return of each asset.
For Asset A:
For Asset B:
Once you know both expected returns, calculate covariance:
Calculate each asset’s variance using the same probability distribution:
Take the square root of each variance to find the standard deviations:
The correlation coefficient is:
Where:
= probability of state
= return on Asset A in state
= return on Asset B in state
= expected return of Asset A
= expected return of Asset B
= deviation of Asset A’s state return from its expected return
= deviation of Asset B’s state return from its expected return
= covariance between the returns of Assets A and B
= variance of Asset A
= variance of Asset B
= standard deviation of Asset A
= standard deviation of Asset B
= correlation coefficient between Assets A and B
= total number of possible states
= the individual state being evaluated
What Do Covariance and Correlation Tell You?
Covariance and correlation both describe how two variables move together, but they present the relationship differently.
Measure | What It Tells You |
|---|---|
Covariance greater than 0 | The returns tend to be above or below their means at the same time |
Covariance less than 0 | One return tends to be above its mean when the other is below |
Covariance equal to 0 | The outcomes show no linear co-movement |
Correlation near +1 | Strong positive linear relationship |
Correlation near -1 | Strong negative linear relationship |
Correlation near 0 | Weak or no linear relationship |
Covariance depends on the units used in the calculation. Returns entered as decimals produce a different covariance value from returns entered as whole percentages.
Correlation removes that unit dependence. Its fixed range makes it easier to compare relationships across different asset pairs.
A correlation near zero only describes the linear relationship. The two variables may still have a nonlinear relationship.
How to Calculate Covariance and Correlation
Use the following order:
Confirm that all probabilities sum to 1.
Calculate .
Calculate .
Find each state return’s deviation from its expected return.
Multiply the paired deviations in each state.
Multiply each cross-product by its state probability.
Add the weighted cross-products to calculate covariance.
Calculate the variance and standard deviation of each asset.
Divide covariance by the product of the standard deviations.
This sequence keeps the inputs organized. In particular, both expected returns must be calculated before you find the deviations used in covariance.
Worked Covariance and Correlation Example
Suppose two assets have the following possible paired returns:
State | Probability | Return on Asset A | Return on Asset B |
|---|---|---|---|
1 | 0.20 | -10% | 15% |
2 | 0.30 | 5% | -2% |
3 | 0.30 | 12% | 10% |
4 | 0.20 | 25% | 18% |
The probabilities sum to 1, so the distribution covers all four possible states.
Step 1: Calculate Asset A’s Expected Return
Asset A’s expected return is 8.1%.
Step 2: Calculate Asset B’s Expected Return
Asset B’s expected return is 9.0%.
Step 3: Calculate Covariance
For each state, subtract the relevant expected return from each asset’s return. Multiply the two deviations, weight the product by the state probability, and then add the results.
State | |
|---|---|
1 | -0.002172 |
2 | 0.001023 |
3 | 0.000117 |
4 | 0.003042 |
Total covariance | 0.002010 |
The covariance is positive. Across the full probability distribution, the two assets tend to move above or below their expected returns together.
Some individual states show returns moving in opposite directions. Covariance summarizes the probability-weighted relationship across all states rather than requiring every paired movement to have the same sign.
Step 4: Calculate the Standard Deviations
For Asset A:
Asset A’s standard deviation is approximately 11.41%.
For Asset B:
Asset B’s standard deviation is approximately 7.75%.
Step 5: Calculate the Correlation Coefficient
The correlation is approximately 0.23.
This result indicates a weak positive linear relationship. The assets generally move in the same direction, but the relationship is not especially strong.
Common Exam Traps
Common mistakes include:
Treating a joint probability as the probability of only one asset’s return.
Forgetting to confirm that all state probabilities sum to 1.
Calculating covariance before finding both expected returns.
Subtracting Asset A’s expected return from Asset B’s state return.
Forgetting to weight each cross-product by its state probability.
Using variance rather than standard deviation in the correlation denominator.
Mixing returns expressed as decimals with returns expressed as whole percentages.
Assuming positive covariance means both assets earn positive returns in every state.
Treating zero covariance as proof that the variables are independent.
Reporting a correlation coefficient outside the range of 1 to +1.
Practice Question
Two equally likely states have the following returns:
State | Probability | Asset X | Asset Y |
|---|---|---|---|
1 | 0.50 | 10% | 5% |
2 | 0.50 | -10% | -5% |
The expected return of each asset is 0%. What is the covariance between Asset X and Asset Y?
0.000
0.005
0.050
Correct Answer: B
The covariance is:
Both cross-products are positive. In the first state, both returns are above their means. In the second state, both are below their means.
Option A would indicate no probability-weighted linear co-movement.
Option C is ten times too large and may result from mixing percentage and decimal units.
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FAQs About Covariance and Correlation
How Do You Calculate Correlation From Covariance?
Divide the covariance between the two variables by the product of their standard deviations.
This calculation standardizes covariance and produces a result between -1 and +1.
What Is the Difference Between Covariance and Correlation?
Covariance shows the direction of the relationship between two variables, but its size depends on the units used. Correlation shows both direction and relative strength on a standardized scale from -1 to +1.
Is Covariance From a Joint Probability Distribution the Same as Sample Covariance?
No. Covariance from a joint probability distribution uses known state probabilities to weight possible paired outcomes.
Sample covariance estimates the relationship from observed sample data. Its calculation commonly uses in the denominator rather than known probability weights.
Can Covariance Be Positive When Some Returns Move in Opposite Directions?
Yes. Covariance summarizes the probability-weighted relationship across every state. Some paired outcomes may move in opposite directions while the overall covariance remains positive.
Does Zero Correlation Mean Two Variables Are Independent?
Not necessarily. Zero correlation indicates no linear relationship. A nonlinear relationship may still exist, so zero correlation alone does not establish independence.